4.1 $BFsJ,K!(B (bisection method)

$BHy@QJ,$G4pK\E*$J(B$BCf4VCM$NDjM}(B$B$rI|=,$7$h$&!#(B

($B$D$^$j(B $f(\alpha)f(\beta)<0$ $B$H$J$k(B $\alpha$, $\beta$ $B$,$"$l$P!"J}Dx(B $B<0(B $f(x)=0$ $B$N2r(B $x=c$ $B$,6h4V(B $(\alpha,\beta)$ $BFb$KB8:_$9$k$H$$$&$3$H!#(B)

$B$3$NDjM}$N>ZL@$N;EJ}$O?'!9$"$k$,!"(B $BBeI=E*$J$b$N$K(B$B6h4V=L>.K!(B$B$r;H$C$?$b$N$,$"$k!#(B $B$=$l$O0J2<$N$h$&$J6Z=q$-$G?J$`!#(B

$B, $\{b_n\}$ $B$rDj$a$k!#(B

(i)

$a_0=\alpha$, $b_0=\beta$ $B$H$9$k!#(B

(ii)

$BBh(B $n$ $B9`(B $a_n$, $b_n$ $B$^$GDj$^$C$?$H$7$F!"(B $c_n=(a_n+b_n)/2$ $B$H$*$-!"(B $f(a_n)f(c_n)<0$ $B$J$i(B $a_{n+1}=a_n$, $b_{n+1}=c_n$, $B$=$&$G$J$$$J$i(B $a_{n+1}=c_n$, $b_{n+1}=b_n$ $B$H$9$k!#(B

$B$9$k$H!"(B

$$a_0 \le a_1 \le a_2 \le \cdots \le a_n \le a_{n+1} \le \cdo... ...\cdots \le b_{n+1} \le b_n \le \cdots \le b_2 \le b_1 \le b_0$$

$$a_n < b_n \le b_0 \quad\hbox{($n\in\N$)}\quad \hbox{$B$5$i$K(B} \quad a_0\le a_n < b_n \quad\hbox{($n\in \N$)},$$

$$b_n-a_n=(\beta-\alpha)/2^n \to 0 \quad \hbox{(as $n\to \infty$)},$$

$$f(a_n)f(b_n)\le 0 \quad\hbox{($n\in\N$)}.$$

$B$3$l$+$i(B

$$\lim_{n\to+\infty}a_n=\lim_{n\to+\infty}b_n=c, \quad \alpha < c < \beta$$

$B$H<}B+$7$F(B

$$f(c)=0$$

$B$,@.$jN)$D$3$H$,J,$+$k!#(B

$B0J>e$N>ZL@$N $f(\alpha)f(\beta)<0$ $B$H$J$k(B $\alpha$, $\beta$ $B$,J,$+$C$F$$$k>l9g$K!"J}Dx<0(B $f(x)=0$ $B$N6a;w2r$r5a$a$k%"%k%4%j(B $B%:%`$,F@$i$l$^$9(B ($B0J2<$G$O(B $\leftarrow$ $B$OJQ?t$X$NBeF~$rI=$7$^$9(B)$B!#(B

[

l]$BFsJ,K!$N%"%k%4%j%:%`(B

1. $BL\I8$H$9$k8m:9(B $\varepsilon$ $B$r7h$a$k!#(B
2. $a \leftarrow \alpha$, $b \leftarrow \beta$ $B$H$9$k!#(B
3. $c \leftarrow (b + a) / 2$ $B$H$7$F(B $f(a)f(c)<0$ $B$J$i$P(B $b \leftarrow c$$B!"$=$&(B $B$G$J$1$l$P(B $a \leftarrow c$ $B$H$9$k(B
4. $\vert b-a\vert \ge \varepsilon$ $B$J$i$P(B (1) $B$KLa$k!#$=$&$G$J$1$l$P(B $c$ $B$r(B $B2r$H$7$F=PNO$9$k!#(B